FIG. 1 depicts a prior art receiver of a Multi-Carrier Modulated (MCM) signal. The MCM signal is sensed by antenna element 100 and fed 102 to module 110. Module 110 contains a Low Noise Amplifier (LNA) 114 and possibly some form of switch diagrammatically depicted as 116 which collectively feed 112 down converter 120.
The transmitted frequency may use any of several frequency bands, 900 MHz, 2-2.5 GHz and 5-6 GHz, being commonly used transmission bands.
Down converter 120 generates a down converted version of the sensed signal, which is fed 122 to Variable Gain Amplifier 124, which generates an amplified, converter signal which is fed 126 to Band Pass Filter 130. Band Pass Filter 130 removes undesirable noise components introduced by the down conversion and amplification, generating a filtered intermediate frequency signal 132.
The filtered intermediate frequency signal 132 is coherently split into two coherent intermediate frequency signals 134 and 136. Intermediate frequency signal 134 is presented to mixer 160 and intermediate frequency signal 136 is presented to mixer 170. Local Oscillator (LO) 140 generates a stable reference signal 142 which is split into two coherent reference signals 144 and 146. Reference signal 144 is presented to mixer 160. Reference signal 146 is presented to phase offset 150, which generates a phase offset reference signal 152 which is presented to mixer 170. Phase offset 150 imparts the equivalent of a phase shift of 90 degrees onto phase offset reference signal 152 with respect to reference signal 146.
Mixer 160 uses intermediate frequency signal 134 and reference signal 144 to create a first intermediate frequency component signal 162 in a frequency range compatible with A/D converter 190.
Mixer 170 uses intermediate frequency signal 136 and phase offset reference signal 152 to create a second intermediate frequency component signal 172 in a frequency range compatible with A/D converter 210.
Signals 162 and 172 may include undesirable frequency components requiring further filtering before presentation to A/D converters 190 and 210, respectively, but these filters have not been shown in the interests of clarity.
Often the first and second intermediate frequency component signals 162 and 172 contain signals in a frequency range under 100 MHz, in many cases on the order of 1-20 MHz. The maximum frequency range is often known as the intermediate frequency and determines the band pass frequency range of band pass filter 130.
A/D converters 190 and 210 respectively generate first sample data stream 192 and second sample data stream 212. The sampling rate is usually a multiple of the intermediate frequency, which by the Nyquist theorem is theoretically 2, and often in practice at least 2.5.
The sampled data streams 192 and 212, also known as the I and Q sample streams, are sent to digital processor 250. In many situations, they are merged, buffered 260, then conveyed 262 across a communication network 270, and delivered 272 to processing engine 280.
The sample sizes of the A/D converters vary for specific applications, but may be any of 6, 7, 8, 9, 10, 12 or more bits per sample. These digitized samples are often packed into a computer format of 8, 16, or 32 bits.
Processing engine 280 will often place these merged digitized samples into at least one sample input buffer 300 residing in memory 290. Memory 290 is accessibly coupled 282 to processing engine 280. Based upon these buffered, digitized samples 300, processing engine 280 will often utilize a heuristic mechanism to determine whether the sampled channel sensed by antenna 100, and processed by the above discussed mechanism is active, generating a Clear Channel Assessment (CCA) flag 310.
The prior art has focused upon Clear Channel Assessment based upon classical signal detection theory. Such developments focus Bayesian detection mechanisms, which calculate the probability of successful detection against the probability of false detection. Such mechanisms tend to require long start-up times, preferably capturing at least one start of burst or message header containing a training sequence, allowing timing synchronization between the user and the transmitting communications medium.
A/D converters 190 and 210 usually sample their respective input signals at the same rate, in fact most often sampling those signals using a carefully constructed clocking scheme controlling clocking skew between them.
One or more clocks may control the timing of processing engine 280. One clock may be operating above 100 MHz, and perhaps operating at much higher rates, such as 240 MHz or higher.
Processing engine 280 may transfer the merged data samples from temporary buffer 260 based upon the triggering of an interrupt, in some cases using a Direct Memory Access (DMA) mechanism (not shown). The DMA mechanism may operate across network 270 to transfer the digitized input samples to sample inputs 300 residing in memory 290. Note that memory 290 may include both volatile and non-volatile memory components.
One or more program counters may control the operations of processing engine 280. Processing engine 280 fetches one instruction for each of the program counters to control the operations of processing engine 280. Note that a single program counter would be compatible with a Single Instruction processing engine, whereas multiple program counters would be compatible with a Multiple Instruction processing engine.
Alternatively, the operation of processing engine 280 may be controlled by a collection of states, such as the one-hot state machines often found in FPGA-based designs. As used herein, a program step will refer to at least an instruction or processor control state providing the controls necessary to execute one or more steps of the inventive method. A program system as used herein will refer to the collection of program steps implementing an embodiment of the inventive method.
Inverse linear transform 340, as found in the prior art, approximates the inverse of a non-singular linear transform which was used to create a signal progression. That signal progression transports across the physical transport layer(s) of the communication protocol to create the transported version of the signal progression received as the sample list in buffer 260 and subsequently found in sample inputs 300.
As used herein, MCM refers to the communication of a data stream by dividing that data stream into multiple parallel sub-streams, each having a lower bit rate, and then concurrently modulating these sub-streams with separate sub-carriers. The separate carriers may or may not be isolated from each other. Frequency Division Multiple Access (FDMA) protocols, including AMPS and GSM, use frequency bins separated by guard bands as the sub-carriers as in FIG. 2A. Such MCM protocols require steep bandpass filters that completely separate the sub-carriers.
Orthogonal Frequency Division Multiplexing (OFDM) uses densely spaced sub-carriers and overlapping spectra as shown in FIG. 2B, eliminating the need for steep bandpass filters. OFDM sub-carriers are, by construction, mutually orthogonal within the protocol specified sampling window. In many cases, both the transmitter and receiver employ complementary Fast Fourier Transforms (FFT) and Inverse Fast Fourier Transforms (IFFT) to transmit and receive the data sub-streams.
OFDM has been studied for use, in conjunction with Direct Sequence Spread Spectrum techniques, to create CDMA-OFDM protocols as shown in FIG. 2C. The transmitter would first apply a Walsh-Hadamard transform to the multiple data sub-streams to create multiple spread data sub-streams. The multiple spread data sub-streams would then be transformed by an IFFT to create an intermediate frequency modulated signal, which is then up-converted to the transmission frequency band.
The receiver would down convert 120 (and filter 130) the amplified antenna reception 112 to create a received intermediate frequency 132 as discussed in FIG. 1 above. This received intermediate frequency signal 132 would be split (143 and 136) and mixed (160 and 170, respectively) with a local reference 144 and a phase offset version of the reference 152, which would be sampled by A/D converters 190 and 210, respectively. The output of A/D converters 190 and 210 are the I and Q sample streams as discussed above.
OFDM protocol research has lead to the specification and deployment of communications protocols in a variety of application areas including, but not limited to, Digital Video Broadcast, Digital Audio Broadcast, and wireless data networks.
The following formulae provide a first of two equivalent definitions of Walsh-Hadamard transforms as used in spread spectrum communications. H1=[0]  (1)                               H          2                =                              [                                                            0                                                  0                                                                              0                                                  1                                                      ]                    =                      [                                                                                H                    1                                                                                        H                    1                                                                                                                    H                    1                                                                                                              H                      1                                        _                                                                        ]                                              (        2        )                                          H          4                =                              [                                                                                H                    2                                                                                        H                    2                                                                                                                    H                    2                                                                                                              H                      2                                        _                                                                        ]                    =                      [                                                            0                                                  0                                                  0                                                  0                                                                              0                                                  1                                                  0                                                  1                                                                              0                                                  0                                                  1                                                  1                                                                              0                                                  1                                                  1                                                  0                                                      ]                                              (        3        )                                          H                      2            ⁢            N                          =                  [                                                                      H                  N                                                                              H                  N                                                                                                      H                  N                                                                                                  H                    N                                    _                                                              ]                                    (        4        )            
The H matrices use an alphabet of two symbols, 0 and 1. In such an alphabet, the complement of 0 is 1 and the complement of 1 is 0. The over bar marks in formulae (2) to (4) refer to taking the component-wise complement of each element of the matrix involved under the bar.
Formula (1) depicts the H1 matrix, which is a 1 by 1 matrix. Formula (2) depicts H2, the 2 by 2 matrix generated as shown from H1. Formula (3) depicts H4, the 4 by 4 matrix generated as shown from H2. Formula (4) depicts H2N the 2N by 2N matrix generated from HN, where N is a power of two.
The following formulae provide a second equivalent definition of Walsh-Hadamard transforms as used in spread spectrum communications. G1=[−1]  (5)                               G          2                =                              [                                                                                -                    1                                                                                        -                    1                                                                                                                    -                    1                                                                    1                                                      ]                    =                      [                                                                                G                    1                                                                                        G                    1                                                                                                                    G                    1                                                                                                              G                      1                                        _                                                                        ]                                              (        6        )                                          G          4                =                              [                                                                                G                    2                                                                                        G                    2                                                                                                                    G                    2                                                                                                              G                      2                                        _                                                                        ]                    =                      [                                                                                -                    1                                                                                        -                    1                                                                                        -                    1                                                                                        -                    1                                                                                                                    -                    1                                                                    1                                                                      -                    1                                                                    1                                                                                                  -                    1                                                                                        -                    1                                                                    1                                                  1                                                                                                  -                    1                                                                    1                                                  1                                                                      -                    1                                                                        ]                                              (        7        )                                          G                      2            ⁢            N                          =                  [                                                                      G                  N                                                                              G                  N                                                                                                      G                  N                                                                                                  G                    N                                    _                                                              ]                                    (        8        )            
Note that various developments of the G matrices may involve a normalization factor, which has not been included. The G matrices use an equivalent alphabet of two symbols, −1 and 1, for which the complement of −1 is 1 and the complement of 1 is −1. The over bar marks in formulae (6) to (8) refer to taking the component-wise complement of each element of the matrix involved under the bar.
Note that the G matrices are more often used in practice, because the absolute value of every entry in the G matrices is the same. The IS-95 communications protocol defines 64 logical channels encoded by G64.
The H matrices are often used for pedagogical purposes or as part of a process leading to the G matrices, providing a more accessible relationship with the standard definitions of bits.
As used herein a bit represents an alphabet possessing two symbols. Multiple bits are the concatenation of single bits, preferably representing a single alphabet.
The prior art also includes Discrete Wavelet Transform (DWT) coding, which is a powerful extension of the linear transform coding discussed to this point.
The following formulae define Discrete Wavelet Transforms as found in the prior art.                     A        =                  [                                                    ⋯                                                              a                                      -                    1                                    0                                                                              a                  0                  0                                                                              a                  1                  0                                                                              a                  2                  0                                                            ⋯                                                                    ⋯                                                              a                                      -                    1                                    0                                                                              a                  0                  1                                                                              a                  1                  1                                                                              a                  2                  1                                                            ⋯                                                                    ⋯                                            ⋯                                            ⋯                                            ⋯                                            ⋯                                            ⋯                                                                    ⋯                                                              a                                      -                    2                                                        m                    -                    1                                                                                                a                  0                                      m                    -                    1                                                                                                a                  1                                      m                    -                    1                                                                                                a                  2                                      m                    -                    1                                                                              ⋯                                              ]                                    (        9        )                                                      ∑                          k              =                              -                ∞                                      ∞                    ⁢                                           ⁢                      a            k            s                          =                  m          ⁢                                           ⁢                      δ                          s              ,              0                                                          (        10        )                                                      ∑                          k              =                              -                ∞                                      ∞                    ⁢                                           ⁢                      a                          k              +              m1                                      ,                              s                ′                                                    ,                                            a                              k                +                m1                            s                        _                    =                      m            ⁢                                                   ⁢                          δ                                                s                  ′                                ·                s                                      ⁢                          δ                              0                ,                1                                                                        (        11        )            Where the Kronecker deltas are defined asif s=s′, then δs,s′=1 else δs,s′=0if l=l′, then δl,l′=1 else δl,l′=0
Formula (9) shows a matrix A with m rows and an unlimited number of columns. A is defined as a wavelet matrix of rank m if formulae (10) and (11) are satisfied by the components of A, ask. These components usually belong to an algebraic sub-field of the complex numbers, such as rational complex numbers, real numbers, rational real numbers, or the complex numbers themselves. To simplify the discussion, the components of A will be assumed to be complex numbers. The over bar of formula (11) refers to taking the complex conjugate of the expression under that bar.                               A          l                =                  [                                                                      a                                      l                    *                    m                                    0                                                                              a                                                            l                      *                      m                                        +                    1                                    0                                                            ⋯                                                              a                                                            l                      *                      m                                        +                    m                    -                    1                                    0                                                                                                      a                                      l                    *                    m                                    1                                                                              a                                                            l                      *                      m                                        +                    1                                    1                                                            ⋯                                                              a                                                            l                      *                      m                                        +                    m                    -                    1                                    1                                                                                    ⋯                                            ⋯                                            ⋯                                            ⋯                                                                                      a                                      l                    *                    m                                                        m                    -                    1                                                                                                a                                                            l                      *                      m                                        +                    1                                                        m                    -                    1                                                                              ⋯                                                              a                                                            l                      *                      m                                        +                    m                    -                    1                                                        m                    -                    1                                                                                ]                                    (        12        )             A=(. . . A−1A0A1 . . . )  (13)
Formula (12) defines a sub-block matrix AI of matrix A containing the columns of A from I*m to I*m+m−1. Formula (13) shows a second way of looking at A as an arbitrary long vector with entries AI. Suppose that only finitely many of the AI components are non-zero. Further suppose that AN1 is the first non-zero component and AN2 is the last non-zero component. Let g=N2−N1+1.
Formulae (14) and (15) define the Laurent series A(z) for the matrix A defined by formula (9) in two different ways.                               A          ⁡                      (            z            )                          ≡                  [                                                                                          ∑                                          k                      =                                              -                        ∞                                                              ∞                                    ⁢                                                                           ⁢                                                            a                      mk                      0                                        ⁢                                          z                      k                                                                                                  ⋯                                                                                  ∑                                          k                      =                                              -                        ∞                                                              ∞                                    ⁢                                                                           ⁢                                                            a                                              mk                        +                        m                        -                        1                                            0                                        ⁢                                          z                      k                                                                                                                          ⋯                                                                                  ∑                                          k                      =                                              -                        ∞                                                              ∞                                    ⁢                                                                           ⁢                                                            a                                              mk                        +                        r                                            s                                        ⁢                                          z                      k                                                                                                  ⋯                                                                                                          ∑                                          k                      =                                              -                        ∞                                                              ∞                                    ⁢                                                                           ⁢                                                            a                      mk                                              m                        -                        1                                                              ⁢                                          z                      k                                                                                                  ⋯                                                                                  ∑                                          k                      =                                              -                        ∞                                                              ∞                                    ⁢                                                                           ⁢                                                            a                                              mk                        +                        m                        -                        1                                                                    m                        -                        1                                                              ⁢                                          z                      k                                                                                                    ]                                    (        14        )                                          A          ⁡                      (            z            )                          =                              ∑                          l              =                              -                ∞                                      ∞                    ⁢                                           ⁢                                    A              l                        ⁢                          z              l                                                          (        15        )            
From hereon, this discussion will assume that only finitely many of the AI components are non-zero. Such matrices A will be known as discrete wavelet matrices. To further simplify the discussion, from hereon N1 will be taken to be zero. As one of skill in the art will realize that N1 could be non-zero and the resulting matrices would essentially be equivalent to “shifted” matrices where N1=0.
Any DWT is equivalent to a matrix A, which has m rows and m*g columns, where m is called the rank and g is the genus of the transform. Each DWT transform is further characterized by an m*m characteristic Haar matrix which equals A(1). When g is one, the transform has a square matrix equal to its Haar characteristic matrix.
Formula (11) asserts that the rows of a wavelet matrix as=(as0, . . . , asmg−1) have length m1/2 and that they are pair-wise orthogonal when shifted by an arbitrary multiple of m.
The vector a0 is often called the scaling vector, low pass filter, or scaling filter. The vectors as for 0<s<m are often called wavelet vectors, high-pass filters, or wavelet filters. Formula (10) states that the sum of the components of the scaling vector is m, whereas for each wavelet vector, the sum of components is zero.
The rank m of a transform corresponds to the sampling rate and to the number of bands in an m-band filter bank implementation. If a filter has rank m, then it samples the signal m times per unit time. When m is infinite, the sampling is continuous and the filter is analog.
The genus g of the transform represents the number of symbols or signaling intervals over which the filter operates. When the rank m is finite, m*g equals the number of taps in each sub band filter. Note that if g is infinite, the filter has infinite duration and is not practicable.
Discussion of the characteristic Haar matrix would entail a digression, which is not central to the invention. Suffice it to say that for a given rank m, the choice of Haar characteristic matrices ranges over a continuous (m−1)^2 dimensional family of matrices.
Increasing the rank of m corresponds to increasing the spectral resolution whereas increasing the genus of a transform corresponds to increasing the overlap of successive transform windows.
Note that Fourier transforms as well as Walsh-Hadamard transforms both can be defined, extended and analyzed by DWT techniques, though this is a topic well outside the range of this invention.
Walsh-Hadamard matrices are a special case of Hadamard matrices. Hadamard matrices are square matrices of rank m containing components whose values are either +1 or −1. Hadamard matrices further satisfy HTH=HHT=ml. Note that a Hadamard matrix may be an n*n matrix of rank m, where m is less than n.
Consider a linear transform from a domain N dimensional space to a range, which is a second N dimensional space. If the linear transform does not cover the range, in other words, if there is a point in the second N dimensional space for which there is no point in the domain which transforms to that point, then the linear transform is singular. If for every point of the second N dimensional space, there is exactly one corresponding point in the first N dimensional space which transforms to that point, then the transform is non-singular.
Linear transforms are well known to have matrices associated with them. When the linear transform is non-singular, its matrix is non-singular, has non-zero determinant and possesses an inverse, which is also non-singular with non-zero determinant.
When the linear transform is singular, its matrix is singular, has determinant 0 and does not possess a non-singular inverse.
A further problem occurs when a linear transform goes from an N dimensional space to an M dimensional space, where N and M are distinct. Again, there is a matrix associated with the transform, but the transform cannot be non-singular.
Work by a number of people, including Moore, Penrose, and Drazin, has led to a theory of inverses applicable to singular matrices, giving rise to several sometimes distinct, pseudo-inverses of a matrix. As may be expected, the pseudo-inverse of a non-singular (necessarily square) matrix is the classic inverse, which is non-singular.
As used herein, R(A) will refer to the range of the linear transform for the associated matrix A and N(A) will refer to the null space of the linear transform for the associated matrix A. The addition of two vector spaces over the same field (which will usually be C, the complex numbers) is the vector space including exactly all linear combinations of the vectors of the two vector spaces. Note that much of this discussion is applicable to vector spaces over algebraic fields in general and in specific, almost always to algebraic sub-fields of the complex numbers including the rational real numbers, real numbers, and rational complex numbers, as well as the complex numbers. The discussion from hereon will focus on vector spaces over the complex number field for convenience and is not intended to restrict the scope of the claims herein.
The conjugate transpose of a matrix A will be denoted herein as A* and will include components for a given row and column which are complex conjugates of the component of the column and row of A.
Given a matrix A of m rows and n columns of m*n complex number components, a matrix G of n rows and m columns of n*m complex number components is called herein an (i,j,k)-inverse of A if G satisfies the ith, jth and kth Penrose conditions:    1. AGA=A    2. GAG=G    3. (AG)*=AG    4. (GA)*=GA
The set of all (l,j,k)-inverses for A will be denoted by A{l,j,k}.
The following are some basic facts about some of the various classes of inverses developed in detail in Generalized Inverses of Linear Transformations by S. L. Campbell and C. D. Meyer, Jr., © 1979, first published by Dover in 1991, ISBN 0-486-66693-X, particularly in chapter 6.
G belongs to A{1} if and only if Qb is a solution of Ax=b, for every vector b in the range of A. This type of inverse is denoted (1)-inverse and is called the Equation Solving Inverse.
If G belongs to A{1,2}, then N(A)+R(G)=Cn and R(A)+N(G)=Cm. Each (1,2)-inverse defines complementary subspaces for N(A) as well as R(A). Conversely, for each pair of subspaces (P,Q), where P and Q are complementary to N(A) and R(A), respectively, uniquely determine a (1,2)-inverse, GP,Q with R(GP,Q)=P and N(GP,Q)=Q. This type of inverse will be called a prescribed range/null space inverse.
Two vector subspaces of a vector space will be referred to as complementary if the only element they have in common is the origin, and if linear combinations of vectors from these two subspaces cover the whole vector space.
G belongs to A{1,3} if and only if Gb is a least squares solution of Ax=b for every vector b in Cm. Gb will be referred to as a least squares solution of Ax=b when the distance between the hyperplane Ax and the vector b is minimal at Gb. This type of inverse will be called a least squares inverse.
G belongs to A{1,4} if and only if Gb is the minimum norm solution of Ax=b for every vector b in R(A). This type of inverse will be called a minimum norm inverse.
As used herein, the norm of a vector b is formed as the square root of the product b and b*. Note that the product of b and b* is a non-negative real number. The minimum norm solution Gb has the least norm of any solution of Ax=b.
A{1,2,3,4} contains exactly one element, denoted as A+ herein. A− is the (R(A*), N(A*))-inverse for A. A+b is the minimal norm least squares solution of Ax=b for any b in Cm. If b belongs to R(A), then A+b is the minimal norm solution of Ax=b. A+ is known elsewhere as the Moore-Penrose Inverse.
Computing the Moore-Penrose inverse A+ from the above definitions involves an unpleasant fact. If A is of neither full row rank nor full column rank, then the rank of A may be perturbed in an arbitrarily small way, dramatically changing the value of A+ (see page 247 of Campbell and Meyer for a discussion and proof).
Another approach to calculating the Moore-Penrose inverse A+ involves use of the Singular Value Decomposition Theorem (see pages 6, and 247-262 of Campbell and Meyer). Matrix A is factored into A=UEV, where U and V are unitary (square) matrices and E has the form                E=[Diag(Eigen(A*A)1/2) 0]                    [0 0]                        
The E matrix includes the positive eigenvalues of the matrix of the square roots of A*A.
Defining A+=V*E+U*, can be shown to provide a well defined pseudo-inverse varying continuously with A regarding suitably chosen matrix norms (pages 247-262 of Campbell and Meyer, matrix norms are defined and discussed on pages 210-213). This definition is often taken as the standard for these reasons.
While much more can be said about this topic, the above definition is computationally demanding. Matrix inverses of non-singular square matrices using Gaussian elimination take on the order of N3 operations. Calculation of all the eigenvalues of a matrix (A*A)1/2  is an even greater task. A number of specialized algorithms are discussed in Campbell and Meyer, as well as in other literature sources which are much faster and often useful, but lack the generality of the Singular Value Decomposition derived A+.
Note that the Moore-Penrose inverse, as well as {i,j,k}-inverses, provide either some form of solution, or least-squares solution, for a linear algebraic system. While possessing many important qualities, these inverses lack some other desirable qualities. Let A and B be n*n complex matrices, there is no class C(i,j,k) of {i,j,k}-inverses A− and B− for A and B respectively which imply any of the following:    1. AA−=A−A,    2. (A−)p=(Ap)− for all positive integers p,    3. λ is an eigenvalue of A if and only if 1/λ is an eigenvalue of A−.    4. Ap+1A−=Ap, for positive integers p.
The Drazin inverse and group inverse have at least these properties.
Before discussing the Drazin inverse, we need to define the index of linear transformation A from Cn to Cn, which will be denoted herein as Ind(A). Ind(A) is the smallest non-negative integer k such that Cn=R(Ak)+N(Ak), where A2=A applied to A, Am+1=A applied to Am, for any positive integer m. Note that if A is non-singular, Ind(A)=0 and that Ind(O)=1. Further, if k=Ind (A), then R(Ak)=R(Ak+1).
There are two ways to define the Drazin inverse of a square matrix A associated with linear transformation A from Cn to Cn.
Let A be a linear transformation on Cn such that k=Ind(A). Let A1=A restricted to R(Ak). Let x=u+v belong to Cn, where u belongs to R(Ak) and v belongs to N(Ak). A1 is invertible and define ADx=A1−1u. AD is the Drazin inverse of linear transformation A. This definition is known as the Functional Definition of the Drazin inverse.
The Algebraic Definition of the Drazin inverse defines AD in Cn*n for A in Cn*n with Ind(A)=k as a matrix satisfying the following:ADAAD=ADAAD=ADA andAk+1AD=Ak.
These definitions are equivalent, and for any A in Cn*n, AD exists and is unique. If A is non-singular, then AD is exactly the standard matrix inverse.
Note that AD is not always a {1}-inverse for A, it doesn't always solve Ax=b. In fact ADb is a solution of Ax=b if and only if b belongs to R(Ak), where k=Ind(A).
In one important case, when Ind(A)<=1, AD solves Ax=b. In such cases AD is called the Group Inverse of A and is often denoted by A#. When it exists, A#can be alternatively defined as the unique matrix satisfyingAA#A=AA#AA#=A#AA#=A#A
Finally, the relationship between the Moore-Penrose inverse A+ and the Drazin inverse AD for a matrix A in Cn*n, A+=AD if and only if A+A=AA+.
Modern radio receivers such as depicted in FIG. 1 often face situations where the active reception channels to be decoded are a subset of all the channels supported by the protocol. Note that these channels are the received version of signals transmitted after being linear transformed by any of a number of matrices. These transform matrices may or may not be square matrices. They may not be invertible, even if they are square matrices.
An example of this is found in the IEEE 802.11a protocol, where there are always 12 null frequency bins, but during the header transmission there are only 12 active data frequency bins out of the 64 frequency bins in the protocol. While only a small part of the frequency bins are required, there is no technique available to describe and control DSP resources at any level finer than a linear transformation. As a consequence, vital computational resources are expended when only a small part of those resources need be used. During the header, only 12 out of the 64 results of the FFT are used. Note that the range of the encoding transformation for the preamble has a dimension of 12 out of 64. The range of the encoding transformation of the body has a dimension of 52 out of 64.
FIG. 2D depicts an example emitted power spectrum requirement for transmitted OFDM signals as found in FIG. 120 of the IEEE 802.11a protocol specification.
“The transmitted spectrum shall have a 0 dBr (dB relative to the maximum spectral density of the signal) bandwidth not exceeding 18 MHz, −20 dBr at 11 MHz frequency offset, −28 dBr at 20 MHz frequency offset and −40 dBr at 30 MHz frequency offset and above. The transmitted spectral density of the transmitted signal shall fall within the spectral mask, as shown in FIG. 120. The measurements shall be made using a 100 kHz resolution bandwidth and a 30 kHz video bandwidth.” (17.3.9.2 Transmit spectrum mask page 28 of the IEEE Standard 802.11a-1999 document)
What is needed is a method of specifying the use of DSP resources based upon inverses of these matrices, which may not necessarily be square matrices, and which may be singular matrices even when they are square matrices. What is further needed is a way to determine the type of inverse of these matrices, as well as calculate the matrix inverse type, most useful for the specific radio reception problem.
Additionally, it is desirable for software radios like that shown in FIG. 1 be able to receive communications in multiple communications protocols. These communications protocols may use the same frequency range for dramatically different protocols. An example of this is the use of the AMPs frequency channels by the IS-95 communications protocol.
IS-95 employs a physical transport layer made of either a single or pair of physical channels reusing the AMPs physical channels. An IS-95 physical channel takes up 41 contiguous AMPs physical channels. When a single IS-95 physical channel is implemented, it is surrounded by a guard band of 9 AMPs physical channels on either side in the frequency domain. When dual IS-95 channels are implemented, the pair of IS-95 physical channels are immediately adjacent to each other, with 9 AMPs physical channels on either side of the pair of IS-95 physical channels.
What is needed is a mechanism to specify linear transformations suitable for rapidly reconfiguring reception by the software radio receiver between distinct communications such as AMPs and IS-95, allowing the reception of channels based upon suitable matrix inverses of these diverse linear transformations.
Cellular telephone users regularly complain about coverage limitations in the United States. A given area will often support some of the wireless protocol standards, which may include AMPs, GSM and IS-95, but not all of them. Most cellular telephones today are built around transceivers, which communicate using only one standard. Cellular telephone transceivers need to sense which protocols are actively supported in the area near which the transceiver is operating.
Cellular telephone users who travel internationally face a very similar problem. Again, most cellular telephones respond to only one of the common standards of today, and there are several distinct standards employed in large areas of the world. These cellular telephones are often useless in areas not supporting that one standard for which they are compatible. Cellular telephone transceivers again need to sense which protocols are actively supported in the area near which the transceiver is operating.